# What are all the possible rational zeros for #f(x)=4x^4+19x^2-63# and how do you find all zeros?

##### 1 Answer

#### Explanation:

By the rational roots theorem, any *rational* zeros of

That means that the only possible *rational* zeros are:

#+-1/4, +-1/2, +-3/4, +-1, +-3/2, +-7/4, +-9/4, +-3, +-7/2, +-9/2, +-21/4, +-7, +-9, +-21/2, +-63/4, +-21, +-63/2, +-63#

We could try each of these in turn, but there are easier ways to find the zeros of

Note that

#x^2 = (-19+-sqrt(19^2-4(4)(-63)))/(2*4)#

#=(-19+-sqrt(361+1008))/8#

#=(-19+-sqrt(1369))/8#

#=(-19+-37)/8#

i.e.

Since the result is rational, we could have found this using an AC method instead, but at least the quadratic formula gives it to us directly.

Hence